We present a unified likelihood ratio-based confidence sequence (CS) for any (self-concordant) generalized linear model (GLM) that is guaranteed to be convex and numerically tight. We show that this is on par or improves upon known CSs for various GLMs, including Gaussian, Bernoulli, and Poisson. In particular, for the first time, our CS for Bernoulli has a $\mathrm{poly}(S)$-free radius where $S$ is the norm of the unknown parameter. Our first technical novelty is its derivation, which utilizes a time-uniform PAC-Bayesian bound with a uniform prior/posterior, despite the latter being a rather unpopular choice for deriving CSs. As a direct application of our new CS, we propose a simple and natural optimistic algorithm called OFUGLB, applicable to any generalized linear bandits (GLB; Filippi et al. (2010)). Our analysis shows that the celebrated optimistic approach simultaneously attains state-of-the-art regrets for various self-concordant (not necessarily bounded) GLBs, and even $\mathrm{poly}(S)$-free for bounded GLBs, including logistic bandits. The regret analysis, our second technical novelty, follows from combining our new CS with a new proof technique that completely avoids the previously widely used self-concordant control lemma (Faury et al., 2020, Lemma 9). Numerically, OFUGLB outperforms or is at par with prior algorithms for logistic bandits.

翻译：本文提出了一种基于似然比的统一置信序列(CS)，适用于任何(自协调)广义线性模型(GLM)，该序列被证明是凸的且在数值上是紧致的。我们证明，该置信序列与已知的各种GLM（包括高斯、伯努利和泊松模型）的置信序列相当或更优。特别地，我们首次为伯努利模型构建了不含$\mathrm{poly}(S)$因子的置信半径，其中$S$是未知参数的范数。我们的第一个技术新颖性在于其推导过程：尽管均匀先验/后验在推导置信序列时并非常用选择，但我们通过结合时间均匀的PAC-Bayesian界与均匀先验/后验实现了这一推导。作为新置信序列的直接应用，我们提出了一种简洁自然的乐观算法OFUGLB，适用于所有广义线性赌博机(GLB; Filippi等人(2010))。分析表明，这一经典的乐观方法能够同时为各类自协调（不一定有界）GLB取得最先进的遗憾界，对于有界GLB（包括逻辑赌博机）甚至能实现不含$\mathrm{poly}(S)$因子的遗憾界。我们的第二个技术新颖性体现在遗憾分析中：通过结合新置信序列与全新的证明技术，完全避免了先前广泛使用的自协调控制引理(Faury等人, 2020, 引理9)。数值实验表明，OFUGLB在逻辑赌博机任务中优于或持平现有算法。